A Numerical Procedure for Position Analysis of a Robotic Structure. Part II: 3C Robotic Arm Illustration

Abstract

The paper considers a robotic arm with stipulated geometry and imposed position. In the first part of the work, we deduced the matrix equation that allows for establishing the required corrections for an initial set of displacement values from the pairs of the robot. With the amended set of parameters, a new set of parameters is obtained, providing the robot a position closer to the desired one. This process can be regarded as a sequence from an iterative process that aims a final position of the robotic arm with predetermined accuracy. The second part of the paper applies the relations deduced in the first part, for a certain case, namely a robotic arm that has three cylindrical pairs in the structure. The straightforward application of the equations obtained in the first part develops an over-constrained system of linear equations that necessitates the use of generalized inverse matrix Moon Penrose. This requires, in turn, the application of several matrix operations upon matrices with 12 rows. Next, the equation from the first part is written for the robotic arm with three cylindrical pairs, resulting in the occurrence of several special matrices. The properties of the special matrices are considered and a new simpler method is proposed for finding the corrections necessary to amend the initial parameters. This method reduces the calculus to a system of six equations.

Appendix

Proposition: if $\theta$ is a skew symmetric matrix ($\theta^{T} = - \theta$), then $H = M\theta M^{T}$ is a skew symmetric matrix, too.

Demonstration:

$$ H^{T} = (M\theta M^{T} )^{T} = (M^{T} )^{T} \theta^{T} M^{T} = M( - \theta )M^{T} = - M\theta M^{T} = - H,\,{\text{q}}.{\text{e}}.{\text{d}}. $$

(A1)

$$ \varvec{Z}(\theta ,s) = \left[ {\begin{array}{*{20}c} {cos\,\theta } & { - sin\,\theta } & 0 & 0 \\ {sin\,\theta } & {cos\,\theta } & 0 & 0 \\ 0 & 0 & 0 & s \\ 0 & 0 & 0 & 1 \\ \end{array} } \right];\,\varvec{X}(\alpha ,a) = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & a \\ 0 & {cos\,\alpha } & { - sin\,\alpha } & 0 \\ 0 & {sin\,\alpha } & {cos\,\alpha } & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right] . $$

(A2)

$$ \varvec{Q}_{\theta } = \left[ {\begin{array}{*{20}c} 0 & { - 1} & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right];\varvec{Q}_{s} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] . $$

(A3)